Let $R$ be a finite commutative ring with unity and $p$ a prime integer.
Suppose that there is a primitive $p$th root of unity $u \in R$ such that $u - 1$ is a unit.
Is it true that $u^i - 1$ is also a unit for all $1 \leq i \leq p - 1$? and how to prove it?
What is always the case is $u-1$ and $u^i-1$ generate the same ideal: $$u^i-1=(u-1)(u^{i-1}+\cdots+u+1)$$ and $$u-1=(u^i-1)(u^{i(j-1)}+\cdots+u^i+1)$$ where $ij\equiv1\pmod p$. If then $u-1$ is a unit, then $(u-1)R=R$ and so $(u^i-1)R=R$. Then $u^i-1$ is a unit.